bk7_7.pdf

MEP Y7 Practice Book A

98

7 Number Patterns andSequencesIn this unit we consider how number patterns arise, how to find particular patternsand finding the formula for a general term in a sequence. Again, this topic is animportant building block in mathematical understanding.

7.1 MultiplesWe start by looking at a sequence formed by taking multiples of a particularnumber. For example,

3, 6, 9, 12, 15, . . . , . . .

which are the multiples of 3.

Example

21 3 4 5 6 87 9 10

1211 13 14 15 16 1817 19 20

2221 23 24 25 26 2827 29 30

3231 33 34 35 36 3837 39 40

4241 43 44 45 46 4847 49 50

5251 53 54 55 56 5857 59 60

6261 63 64 65 66 6867 69 70

7271 73 74 75 76 7877 79 80

8281 83 84 85 86 8887 89 90

9291 93 94 95 96 9897 99 100

This square shows the multiples of a number. What is this number?

Write down the numbers that should go in each of these boxes. The numbersquare will help you with some of them.

(a) The 5th multiple of is .

(b) The th multiple of is 36.


99

(c) The 12th multiple of is .

(d) The 20th multiple of is .

(e) The th multiple of is 96.

(f) The 100th multiple of is .

Solution

The number is 4, and

(a) the 5th multiple of 4 is 20,

(b) the 9th multiple of 4 is 36,

(c) the 12th multiple of 4 is 48,

(d) the 20th multiple of 4 is 80,

(e) the 24th multiple of 4 is 96,

(f) the 100th multiple of 4 is 400.

Exercises1. On a number square like this one, shade all the multiples of 6. Then answer

the questions.

21 3 4 5 6 87 9 10

1211 13 14 15 16 1817 19 20

2221 23 24 25 26 2827 29 30

3231 33 34 35 36 3837 39 40

4241 43 44 45 46 4847 49 50

5251 53 54 55 56 5857 59 60

6261 63 64 65 66 6867 69 70

7271 73 74 75 76 7877 79 80

8281 83 84 85 86 8887 89 90

9291 93 94 95 96 9897 99 100


100

(a) What is the 4th multiple of 6?

(b) What is the 10th multiple of 6?

(c) What is the 12th multiple of 6?

(d) What is the 100th multiple of 6?

2. The multiples of a number have been shaded on this number square. Whatis the number?

21 3 4 5 6 87 9 10

1211 13 14 15 16 1817 19 20

2221 23 24 25 26 2827 29 30

3231 33 34 35 36 3837 39 40

4241 43 44 45 46 4847 49 50

5251 53 54 55 56 5857 59 60

6261 63 64 65 66 6867 69 70

7271 73 74 75 76 7877 79 80

8281 83 84 85 86 8887 89 90

9291 93 94 95 96 9897 99 100

Copy each statement about these multiples and write down the numbers thatshould go in the boxes.

(a) The 3rd multiple of is .

(b) The 9th multiple of is .

(c) The 200th multiple of is .

(d) The th multiple of is 66.

(e) The th multiple of is 330.

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101

3. (a) Write down the first 8 multiples of 8.

(b) Write down the first 8 multiples of 6.

(c) What is the smallest number that is a multiple of both 6 and 8?

(d) What are the next two numbers that are multiples of both 6 and 8?


(b) What is the 10th multiple of 12?

(c) What is the 100th multiple of 12?

(d) What is the 500th multiple of 12?

(e) If 48 is the nth multiple of 12, what is n?

(f) If 96 is the nth multiple of 12, what is n?

5. (a) What multiples have been shaded in this number square?

(b) What is the first multiple not shown in the number square?

6. (a) Explain why 12 is a multiple of 6 and 4.

(b) Is 12 a multiple of any other numbers?

7. The number 24 is a multiple of 2 and a multiple of 3. What other numbersis it a multiple of?

21 3 4 5 6 87 9 10

1211 13 14 15 16 1817 19 20

2221 23 24 25 26 2827 29 30

3231 33 34 35 36 3837 39 40

4241 43 44 45 46 4847 49 50

5251 53 54 55 56 5857 59 60

6261 63 64 65 66 6867 69 70

7271 73 74 75 76 7877 79 80

8281 83 84 85 86 8887 89 90

9291 93 94 95 96 9897 99 100


102

8. Two multiples of a number have been shaded on this number square. Whatis the number?

9. Two multiples of a number have been shaded on this number square.

21 3 4 5 6 87 9 10

1211 13 14 15 16 1817 19 20

2221 23 24 25 26 2827 29 30

3231 33 34 35 36 3837 39 40

4241 43 44 45 46 4847 49 50

5251 53 54 55 56 5857 59 60

6261 63 64 65 66 6867 69 70

7271 73 74 75 76 7877 79 80

8281 83 84 85 86 8887 89 90

9291 93 94 95 96 9897 99 100

(a) What is the number?

(b) What is the 19th multiple of this number?

21 3 4 5 6 87 9 10

1211 13 14 15 16 1817 19 20

2221 23 24 25 26 2827 29 30

3231 33 34 35 36 3837 39 40

4241 43 44 45 46 4847 49 50

5251 53 54 55 56 5857 59 60

6261 63 64 65 66 6867 69 70

7271 73 74 75 76 7877 79 80

8281 83 84 85 86 8887 89 90

9291 93 94 95 96 9897 99 100

7.1


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10. Three multiples of a number are 34, 170 and 255. What is the number.

11. Three multiples of a number are 38, 95 and 133. What is the number?

12. Four multiples of a number are 49, 77, 133 and 203. What is the number?

7.2 Finding the Next TermHere we use the given numbers of the sequence to deduce the pattern and hencefind the next term

Example

What are the next 3 numbers in the sequences:

(a) 12, 17, 22, . . .

(b) 50, 47, 44, 41, 38, . . .

Solution

(a) To spot the pattern, it is usually helpful to first find the differences betweeneach term; i.e.

12 17 22

5 5

So the next term is found by adding 5 to the previous term; this gives 27,32, 37.

(b) Again we find the difference:

50 47 44 41 38

–3 –3 –3 –3

So the next term is found by taking away 3 from the previous term, giving35, 32, 29.


104

Exercises1. Copy each of the sequences below and write in the next 3 numbers in each

sequence. Complete the working that is shown.

(a) 1, 4, 7, 10, 13, . . .

3 3 3

(b) 3, 5, 7, 9, 11, . . .

(c) 5, 8, 11, 14, 17, . . .

(d) 6, 8, 10, 12, 14, . . .

(e) 20, 19, 18, 17, 16, . . .

(f) 6, 9, 12, 15, 18, . . .

(g) 22, 20, 18, 16, 14, . . .

2. Copy each sequence and fill in the missing number.

(a) 4, 7, , 13, 16, . . .

(b) 7, , 15, 19, 23, . . .

(c) 8, 14, 20, , 32, . . .

(d) 3, 11, , 27, 35, . . .

(e) 15, , 27, 33, 39, . . .

3. Copy and continue each sequence, giving the next three numbers.

(a) 18, 30, 42, 54, 66, . . .

(b) 4.1, 4.7, 5.3, 5.9, 6.5, . . .

(c) 14, 31, 48, 65, 82, . . .

(d) 101, 119, 137, 155, 173, . . .

(e) 3.42, 3.56, 3.70, 3.84, 3.98, . . .

(f) 10, 9.5, 9, 8.5, 8, 7.5, . . .

(g)1

4,

1

2,

3

4, 1, 1

1

4, 1

1

2, . . .

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4. For each sequence of patterns, draw the next two shapes and find the next3 numbers in the sequence.

(a)

6, 12, 20, . . .

(b)

3, 6, 10, 15, . . .

(c)

5, 8, 11, 14, . . .

(d)

4, 9, 16, 25, . . .

5. Find the first number in each of the sequences.

(a) , 6, 11, 16, 21, . . .

(b) , 7, 9, 11, 13, . . .

(c) , 6, 5, 4, 3, . . .

(d) , 19, 28, 37, 46, . . .

(e) , 12, 9, 6, 3, . . .


106

6. Copy each sequence and write in the next three terms.

(a) 1, 4, 9, 16, 25, . . .

(b) 2, 5, 10, 17, 26, . . .

(c) 0, 3, 7, 12, 18, . . .

(d) 6, 12, 20, 30, 42,

(e) 0.5, 2.0, 4.5, 8.0, 12.5, . . .

7. Copy each sequence and fill in the missing numbers.

(a) 2, 4, , 16, 32, . . .

(b) 100, 81, 64, , 36, . . .

(c) 6, 9, , 21, 30, . . .

(d) 0, 1.5, 4, , 12, . . .

(e) 1, 7, 17, , 49, . . .

8. Write down the next two terms in each sequence.

(a)1

2

2

3

3

4

4

5, , , , . . .

(b)9

11

8

12

7

13

6

14, , , , . . .

(c)3

6

5

7

7

8

9

9, , , , . . .

(d)2

1

3

4

4

9

5

16, , , , . . .

(e)0

2

3

5

8

10, , , ,

15

17 . . .

7.3 Generating Number SequencesHere we introduce the concept of the general terms of a sequence. For example,the formula 5n , with n = 1 2 3 4, , , , . . ., generates the sequence

5 1× , 5 2× , 5 3× , 5 4× , . . .

that is 5, 10, 15, 20, . . .

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107

Similarly, 5 1n + gives, in the same way,

5 1 1 5 2 1 5 3 1 5 4 1×( ) + ×( ) + ×( ) + ×( ) +, , , , . . .

that is 6, 11, 16, 21, . . .

Example 1

What sequence is generated by the formulae

(a) 5 1n − (b) 6 2n + ?

Solution

(a) Putting n = 1 2 3 4, , , , . . . gives

4, 9, 14, 19, . . .

(b) Putting n = 1 2 3 4, , , , . . . gives

8, 14, 20, 26, . . .

Example 2

What is the formula for this sequence

11, 21, 31, 41, 51, 61 ?

Solution

As you are starting with 11, and 11 10 1= + , and you continue to add 10 eachtime, the formula will be

10 1n +

Exercises1. What number comes out of each of these number machines?

(a) (b) (c)8

– 5 ?

14

+ 9 ?

7

+ 2 ?


108

(d) (e) (f)

2. What number was put into each of these number machines?

(a) (b) (c)

(d) (e) (f)

3. The sequence 1, 2, 3, 4, 5, . . . is put into each of these machines. Writedown the first 5 terms of the sequence that comes out of each machine.

(a) (b) (c)

Which machine gives the even numbers?


(b) What happens if you put these multiples of 2 into this machine?

7

3 ?×

6

5 ?×

18

– 3 ?

?

+ 1 6

?

+ 2 8

?

– 2 10

?

2 10×

?

+ 5 7

?

– 6 3

2× + 2+ 1

+ 2

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109

5. The sequence 1, 2, 3, 4, 5, . . . is put into each number machine. Whatdoes each machine do?

(a)

(b)

(c)

(d)

6. Write down the first 5 terms of the sequence given by each of theseformulae:

(a) 3n (b) 3 1n + (c) 3 4n +

(d) 3 1n − (e) 3 2n −

7. Write down the first 5 terms of the sequence given by each of theseformulae:

(a) 9n (b) 12n (c) 2 1n +

(d) 3 2n + (e) 5 2n − (f) 7 1n −

1, 2, 3, 4, 5, ...

? 7, 14, 21, 28, 35, ...

1, 2, 3, 4, 5, ...

? 11, 12, 13, 14, 15, ...

1, 2, 3, 4, 5, ...

? 10, 20, 30, 40, 50, ...

1, 2, 3, 4, 5, ...

? 2.5, 3.5, 4.5, 5.5, 6.5, ...


110

8. (a) What is the 10th term of the sequence 2 1n + ?

(b) What is the 8th term of the sequence 3 6n + ?

(c) What is the 5th term of the sequence 4 1n + ?

(d) What is the 7th term of the sequence 5 1n − ?

9. When the sequence 1, 2, 3, 4, 5, . . . is put into the machine:

+ 7

it creates the sequence with formula n + 7.

(a) Write down the first 6 terms of the sequence with formula n + 7.

(b) What happens if the sequence 1, 2, 3, 4, 5, . . . is put into thesemachines? Write down the formula for the sequence you get.

(i) (ii) (iii)

10. You need this machine to get the sequence with formula 4n from thesequence 1, 2, 3, 4, 5, . . .

4×

(a) Write down the first 5 terms of this sequence.

(b) Draw the machine you would need to get6, 12, 18, 24, 30 from 1, 2, 3, 4, 5, . . .

(c) Draw the machine you would need to get the sequence with formula7n from 1, 2, 3, 4, 5, ...

+ 5 – 1 + 2

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111

11. Two machines can be put together like this, to make a double machine.

2 +1×

(a) What do you get if you put 1, 2, 3, 4, 5, . . . into this doublemachine? What is the formula for the sequence you get?

(b) Repeat part (a) for each of these double machines.

(i)

(ii)

(iii)

(iv)

(c) What single machine has the same effect as the double machine inpart (b)(iv)? What is the formula for the single machine?

5 – 2×

+ 3 – 1

2 –1×

4 + 3×


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12. Draw double machines that could be used to get each of these sequencesfrom 1, 2, 3, 4, 5, . . .

Also write down the formula for each sequence.

(a) 5, 9, 13, 17, 21, . . .

(b) 2, 5, 8, 11, 14, . . .

(c) 6, 11, 16, 21, 26, . . .

(d) 4, 9, 14, 19, 24, . . .

(e) 102, 202, 302, 402, 502, . . .

7.4 Formulae for General TermsIt is very helpful not only to be able to write down the next few terms in asequence, but also to be able to write down, for example, the 100th or even the1000th term!

Example

For the sequence

3, 7, 11, 15, . . . , . . .

find (a) the next three terms,

(b) the 100th term,

(c) the 1000th term.

Solution

(a) Looking at the differences,

3, 7, 11, 15

4 4 4

we can see that 4 is added each time to get the next term.

So we obtain 19, 23, 27.

(b) To find the 100th term, starting at 3 (the first term), you add on 4, ninety-nine times, giving

3 4 99+ × = 3 396+

= 399

(c) Similarly, the 1000th term is

3 4 999+ × = 3 3996+ = 3999

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Note

You can go one stage further and write down the formula for a general term, i.e.the nth term.

This is

3 4 1+ × −( )n = 3 4 4+ −n

= 4 1n − (Check the previous answers.)

Exercises1. For the sequence:

2, 5, 8, 11, 14, . . .

(a) What is the difference between each term?

(b) Explain why the formula for the nth term is 3 1n − .

2. For the sequence

6, 8, 10, 12, 14, . . .

(a) find the difference between each term,

(b) explain why the formula for the nth term is 2 4n + .

3. For each sequence, write down the difference between each term and theformula for the nth term.

(a) 3, 5, 7, 9, 11, . . .

(b) 5, 11, 17, 23, 29, . . .

(c) 4, 7, 10, 13, 16, . . .

(d) 2, 5, 8, 11, 14, . . .

(e) 6, 10, 14, 18, 22, . . .

4. (a) What formula gives the sequence

4, 8, 12, 16, 20, . . .

(b) What formula gives the sequence that is the multiples of 5?

5. (a) What is the formula for the nth term of this sequence?

7, 14, 21, 28, 35, . . .

(b) How can you get this sequence from the sequence in (a)?

8, 15, 22, 29, 36, . . .

(c) What is the formula for the nth term of the sequence in (b)?


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(b) What is the formula for the nth term of the sequence of the multiplesof 11?

(c) What is the formula for the nth term of this sequence?

10, 21, 32, 43, 54, . . .

7. Write down the formula for the nth term of each of these sequences.

(a) 3, 6, 9, 12, 15, . . .

(b) 5, 12, 19, 26, 33, . . .

(c) 21, 29, 37, 45, 53, . . .

(d) 8, 11, 14, 17, 20, . . .

(e) 1, 4, 7, 10, 13, . . .

(f) 103, 106, 109, 112, 115, . . .

8. (a) Explain why the formula for the nth term of this sequence,

12

14

16

18

110

, , , , , . . .

is 1

2n.

(b) What is the formula for the nth term of this sequence?

13

15

17

19

111

, , , , , . . .

9. Find formulae for the nth term of each of these sequences.

(a)12

23

34

45

56

, , , , , . . .

(b)14

58

, , , , , 25

36

47

. . .

(c)1

105

14, , , , ,

211

3

12

413

. . .

(d)28

49

610

811

1012

, , , , , . . .

(e)35

66

97

128

159

, , , , , . . .

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10. The formula for the nth term of this sequence is n2 .

1, 4, 9, 16, 25, . . .

What is the formula for the nth term of the following sequences?

(a) 0, 3, 8, 15, 24, . . .

(b) 10, 13, 18, 25, 34, . . .

(c) 2, 8, 18, 32, 50, . . .

(d) 1, 8, 27, 64, 125, . . .

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